78 lines
4.0 KiB
Markdown
78 lines
4.0 KiB
Markdown
An **identity** is an equation that is true for all values of the variable for which the expressions in the equation are defined.
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# Trigonometric Identities
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All of the following only apply when the denominator is not equal to zero.
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$$ tan \theta = \frac{y}{x} $$
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Because the following are inverses of their counterparts, you only need to remember the equivalents for $sin$, $cos$, and $tan$, then just find the inverse by taking $1/v$.
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| Base Identity | Inverse Identity | Alternate Identities | Alternate Inverse Identities |
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| ----------------------------- | ------------------------------ | --------------------------------------------- | --------------------------------------------------------------------- |
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| $$ sin\theta = y $$ | $$ csc\theta = \frac{1}{y} $$ | | $$ csc\theta = \frac{1}{sin\theta} $$ |
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| $$ cos\theta = x $$ | $$ sec \theta = \frac{1}{x} $$ | | $$ sec\theta = \frac{1}{cos\theta} $$ |
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| $$ tan\theta = \frac{y}{x} $$ | $$ cot\theta = \frac{x}{y} $$ | $$ tan\theta = \frac{sin\theta}{cos\theta} $$ | $$ cot\theta = \frac{1}{tan\theta} = \frac{cos\theta}{sin{\theta}} $$ |
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# Pythagorean Identities
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The Pythagorean identity expresses the Pythagorean theorem in terms of trigonometric functions. It's a basic relation between the sine and cosine functions.
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$$ sin^2 \theta + cos^2 \theta = 1 $$
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There are more forms that are useful, but they can be derived from the above formula:
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$$ 1 + tan^2\theta = sec^2\theta $$
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$$ cot^2 \theta + 1 = csc^2\theta $$
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# Even and Odd Identities
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- A function is even if $f(-x) = f(x)$.
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- A function is odd if $f(-x) = -f(x)$
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- Cosine and secant are **even**
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- Sine, tangent, cosecant, and cotangent are **odd**.
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## Examples
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### Even and Odd Functions
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> If $cot\theta = -\sqrt{3}$, what is $cot(-\theta)$?
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$cot$ is an odd function, and so $cot(-\theta) = \sqrt{3}$
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### Simplifying Using Identities
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> Simplify $\frac{sin\theta}{cos\theta}$
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1. The above equation can be split into two components
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$$ \frac{sin\theta}{cos\theta} = \frac{sin\theta}{1} * \frac{1}{csc\theta} $$
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2. Referring to the list of trig identities, we know that $\frac{1}{csc\theta}$ is equal to $sin\theta$.
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$$ \frac{sin\theta}{1} * \frac{1}{csc\theta} = sin\theta * sin\theta $$
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3. Simplifying further, we get:
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$$ sin^2\theta $$
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### Finding all values using identities
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If $sec\theta = -\frac{25}{7}$ and $0 < \theta < \pi$, find the values of the other 5 trig functions:
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1. To find $tan\theta$, we can use the trig identity $1 + tan^2\theta = sec^2\theta$:
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$$ 1 + tan^2\theta = (-\frac{25}{7})^2 $$
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Shuffling things around, we get this:
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$$ tan^2\theta = \frac{625}{49} - 1 $$
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Performing that subtraction gives us this:
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$$ \frac{625}{49} - \frac{49}{49} = \frac{576}{49} = tan^2\theta $$
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You can get rid of the exponent:
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$$ \sqrt{\frac{576}{49}} = tan\theta $$
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$\sqrt{576} = 24$ and $\sqrt{49} = 7$, so:
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$$ tan\theta = \frac{24}{7} $$
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2. To find $cos\theta$, because $sec$ is the inverse of $cos$, we can use the identity $sec\theta = \frac{1}{cos\theta}$:
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$cos\theta = -\frac{7}{25}$
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3. To find $sin\theta$, we can use the trig identity $sin^2\theta + cos^2\theta = 1$:
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$$ sin^2\theta + (-\frac{7}{25}) = 1 $$
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Rearranging, we get:
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$$ 1 - (-\frac{7}{25})^2 = sin^2\theta $$
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Applying the exponent gives us $\frac{49}{625}$, so we can do this:
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$$ \frac{625}{625} - \frac{49}{625} = \frac{576}{625} = sin^2\theta $$
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Getting rid of the exponent:
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$$ \sqrt{\frac{576}{625}} = \frac{24}{25} = sin\theta $$
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From there, you can find the rest of the identities fairly easily.
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# Simplifying trig expressions using identities
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Given the difference of square formula:
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$$ a^2 - b^2 = (a-b)(a+b) $$
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## Examples
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Simplify $\tan\theta\sin\theta + \cos\theta$:
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1. $\dfrac{\sin\theta}{\cos\theta} * \sin\theta + \cos\theta$
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2. $\dfrac{\sin^2\theta}{cos\theta} + \cos\theta$
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3. $\dfrac{1 -\cos^2\theta}{cos\theta} + \cos\theta$
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4. |